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Perimeter Institute Quantum Discussions

This series consists of weekly discussion sessions on foundations of quantum Theory and quantum information theory. The sessions start with an informal exposition of an interesting topic, research result or important question in the field. Everyone is strongly encouraged to participate with questions and comments.

Joint work with Earl Campbell (FU-Berlin) and Hussain Anwar (UCL) Magic state distillation is a key component of some high-threshold schemes for fault-tolerant quantum computation [1], [2]. Proposed by Bravyi and Kitaev [3] (and implicitly by Knill [4]), and improved by Reichardt [4], Magic State Distillation is a method to broaden the vocabulary of a fault-tolerant computational model, from a limited set of gates (e.g.

We study the problem of reconstructing an unknown matrix M, of rank r and dimension d, using O(rd poly log d) Pauli measurements. This has applications to compressed sensing methods for quantum state tomography. We give a solution to this problem based on the restricted isometry property (RIP), which improves on previous results using dual certificates. In particular, we show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r RIP.

Abstract The magic state model of quantum computation gives a recipe for universal quantum computation using perfect Clifford operations and repeat preparations of a noisy ancilla state. It is an open problem to determine which ancilla states enable universal quantum computation in this model. Here we show that for systems of odd dimension a necessary condition for a state to enable universal quantum computation is that it have negative representation in a particular quasi-probability representation which is a discrete analogue to the Wigner function.

I will first present a theorem based on the Decoupling Theorem of [1] which gives sufficient and necessary conditions for a quantum channel (CPTPM) being such that it yields the same output for almost all possible inputs. This theorem allows us to reproduce and generalize results oft [2,3], in which cornerstones of statistical physics are derived from first principles of quantum mechanics, in a very natural and easy way. Specifically, we express them in a way which allows to apply results about random 2-qubit interactions [4].

We provide a microscopic understanding of the nucleation of topological quantum liquids that arise due to interactions between non-Abelian anyons. With the pairwise anyon interactions typically showing RKKY-type oscillations in sign, but decaying exponentially with distance, we show that the character of the nucleated phase is fully determined by anyon interactions beyond nearest neighbor exchange. We investigate this issue in the context of Kitaev's honeycomb lattice model.

We construct a class of entangled supersymmetric states which is used as a non-local resource in the CHSH game. This class of super entangled states is more non-local then maximally entangled states if the supersymmetric degrees of freedom are accessible to measurement.
Consequently, we show that the winning probability for the CHSH game is greater than cos2(pi/8) corresponding to an expected value greater than Tsirelson's bound.

Continuous-variable SICPOVMS seem unlikely to exist, for a variety of reasons. But that doesn't rule out the possibility of other 2-designs for the continuous-variable Hilbert space L2(R). In particular, it would be nice if the coherent states -- which form a rather nice 1-design -- could be generalized in some way to get a 2-design comprising *Gaussian* states. So the question is: "Can we build a 2-design out of Gaussian states?". The answer is "No, but in a very surprising way!" Like coherent states, Gaussian states have a natural transitive symmetry group.

This talk presents sufficient conditions for equilibration and thermalization of subsystems within closed many body quantum systems. That is, we identify when the local properties of the equilibrium state resemble those of a thermal state. With this aim, the recent progress in this field is reviewed and we introduce a novel perturbation technique for a realistic weak coupling between the subsystem and its environment. Unlike the standard perturbation theory, our technique is robust in the thermodynamic limit.

One of the key features of the quantum Hall effect (QHE) is the fractional charge and statistics of quasiparticles. Fractionally charged anyons accumulate non-trivial phases when they encircle each other. In some QHE systems an unusual type of particles, called non-Abelian anyons, is expected to exist. When one non-Abelian particle makes a circle around another anyon this changes not only the phase but even the direction of the quantum-state vector in the Hilbert space. This property makes non-Abelian anyons promising for fault-tolerant quantum computation.

In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated to their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using Projected Entangled Pair States (PEPS). This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian.